In Ref. [1]}, optimal pulses for the robust control of a two-level system haven been successfully obtained by the use of a gradient descent algorithm in the reduced control landscape. In our case however, such an optimization proved to be unsuccessful without a good initial guess for the \(\mathinner {|{\chi _q(0)}\rangle }\) , because small variations in the initial costates can lead to large variations in the final states. Therefore, we introduce the following new procedure to obtain a good initial guess of the \(\mathinner {|{\chi _q(0)}\rangle }\)  based on the pulses found by GRAPE: Denote by \(U_q(t)\)  the time evolution operator under \(H_q\) , so that \(\mathinner {|{\chi _q(t)}\rangle } = U_q(t)\mathinner {|{\chi _q(0)}\rangle }\) . Equation (REF ) then implies that for all \(t\)
\(0 &= \frac{\mathrm {d}}{\mathrm {d}\varphi }\sum _q \mathrm {Im}(\mathinner {\langle {\chi _q(t)| H_q(\varphi ) |\psi _q(t)}\rangle })\\ \nonumber &= \sum _q \mathrm {Im}(\mathinner {\langle {\chi _q(0)|\alpha _q(t)}\rangle })\) 
